Deformation quantization of the $n$-tuple point

Abstract

Contrary to the classical methods of quantum mechanics, the deformation quantization can be carried out on phase spaces which are not even topological manifolds. In particular, the Moyal star product gives rise to a canonical functor $F$ from the category of affine analytic spaces to the category of associative (in general, non-commutative) $\C$-algebras. Curiously, if $X$ is the $n$-tuple point, $x^{n}=0$, then $F(X)$ is the algebra of $n\times n$ matrices.